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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupequzmptf | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupequzmptf.j | ⊢ Ⅎ𝑗𝜑 |
limsupequzmptf.o | ⊢ Ⅎ𝑗𝐴 |
limsupequzmptf.p | ⊢ Ⅎ𝑗𝐵 |
limsupequzmptf.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
limsupequzmptf.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
limsupequzmptf.a | ⊢ 𝐴 = (ℤ≥‘𝑀) |
limsupequzmptf.b | ⊢ 𝐵 = (ℤ≥‘𝑁) |
limsupequzmptf.c | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) |
limsupequzmptf.d | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
limsupequzmptf | ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1918 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | limsupequzmptf.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | limsupequzmptf.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
4 | limsupequzmptf.a | . . 3 ⊢ 𝐴 = (ℤ≥‘𝑀) | |
5 | limsupequzmptf.b | . . 3 ⊢ 𝐵 = (ℤ≥‘𝑁) | |
6 | limsupequzmptf.j | . . . . . 6 ⊢ Ⅎ𝑗𝜑 | |
7 | limsupequzmptf.o | . . . . . . 7 ⊢ Ⅎ𝑗𝐴 | |
8 | 7 | nfcri 2891 | . . . . . 6 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐴 |
9 | 6, 8 | nfan 1903 | . . . . 5 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝐴) |
10 | nfcsb1v 3917 | . . . . . 6 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐶 | |
11 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑗𝑉 | |
12 | 10, 11 | nfel 2918 | . . . . 5 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉 |
13 | 9, 12 | nfim 1900 | . . . 4 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉) |
14 | eleq1w 2817 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴)) | |
15 | 14 | anbi2d 630 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝐴) ↔ (𝜑 ∧ 𝑘 ∈ 𝐴))) |
16 | csbeq1a 3906 | . . . . . 6 ⊢ (𝑗 = 𝑘 → 𝐶 = ⦋𝑘 / 𝑗⦌𝐶) | |
17 | 16 | eleq1d 2819 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝐶 ∈ 𝑉 ↔ ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉)) |
18 | 15, 17 | imbi12d 345 | . . . 4 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉))) |
19 | limsupequzmptf.c | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) | |
20 | 13, 18, 19 | chvarfv 2234 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉) |
21 | limsupequzmptf.p | . . . . . . 7 ⊢ Ⅎ𝑗𝐵 | |
22 | 21 | nfcri 2891 | . . . . . 6 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐵 |
23 | 6, 22 | nfan 1903 | . . . . 5 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝐵) |
24 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑗𝑊 | |
25 | 10, 24 | nfel 2918 | . . . . 5 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊 |
26 | 23, 25 | nfim 1900 | . . . 4 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝐵) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊) |
27 | eleq1w 2817 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐵 ↔ 𝑘 ∈ 𝐵)) | |
28 | 27 | anbi2d 630 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝐵) ↔ (𝜑 ∧ 𝑘 ∈ 𝐵))) |
29 | 16 | eleq1d 2819 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝐶 ∈ 𝑊 ↔ ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊)) |
30 | 28, 29 | imbi12d 345 | . . . 4 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊))) |
31 | limsupequzmptf.d | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) | |
32 | 26, 30, 31 | chvarfv 2234 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊) |
33 | 1, 2, 3, 4, 5, 20, 32 | limsupequzmpt 44380 | . 2 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐶)) = (lim sup‘(𝑘 ∈ 𝐵 ↦ ⦋𝑘 / 𝑗⦌𝐶))) |
34 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
35 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘𝐶 | |
36 | 7, 34, 35, 10, 16 | cbvmptf 5256 | . . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐶) |
37 | 36 | fveq2i 6891 | . . 3 ⊢ (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐶)) |
38 | 37 | a1i 11 | . 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐶))) |
39 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘𝐵 | |
40 | 21, 39, 35, 10, 16 | cbvmptf 5256 | . . . 4 ⊢ (𝑗 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ ⦋𝑘 / 𝑗⦌𝐶) |
41 | 40 | fveq2i 6891 | . . 3 ⊢ (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)) = (lim sup‘(𝑘 ∈ 𝐵 ↦ ⦋𝑘 / 𝑗⦌𝐶)) |
42 | 41 | a1i 11 | . 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)) = (lim sup‘(𝑘 ∈ 𝐵 ↦ ⦋𝑘 / 𝑗⦌𝐶))) |
43 | 33, 38, 42 | 3eqtr4d 2783 | 1 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ⦋csb 3892 ↦ cmpt 5230 ‘cfv 6540 ℤcz 12554 ℤ≥cuz 12818 lim supclsp 15410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-ico 13326 df-limsup 15411 |
This theorem is referenced by: (None) |
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